Remark 1.21 Note that the growth exponent \(\varepsilon \)  in Theorem REF  is independent of \(P\) , and in particular independent of the degree of \(P\)  when \(P\)  is a polynomial. This is interesting for the following reasons. First, if \(P\)  is a polynomial then it might be possible (though likely difficult) to prove an analogue of Theorem REF  using Bourgain's discretized sum-product theorem (Theorem REF ) and tools from Ruzsa calculus. For example, these ideas were used in [1]} to prove expansion over \(\mathbb {F}_p\)  for a certain explicitly specified polynomial. However, the entropy growth exponent \(\varepsilon \)  arising from such a strategy would necessarily depend on the polynomial \(P\)  (the degree of \(P\) , the number of monomial terms, etc).
